In wireless networks, the transmitted signal may suffer high levels of attenuation as a result of the fading phenomenon.
In order to combat this phenomenon, one known technique consists in transmitting a plurality of copies of the same signal in order to create transmission diversity, thereby making it possible to combat the harmful effects of fading effectively. The diversity may be spatial; it is obtained by transmitting signals from different locations. The destination then receives versions of the signal that have been subjected to fading in independent manners.
One of the transmission techniques that enable advantage to be taken of spatial diversity is space-time coding. In a transmission system having a plurality of transmit and/or receive antennas, the use of a space-time code makes it possible to perform space multiplexing (between antennas) and time multiplexing of the information symbols to be transmitted so as to take best advantage of the degrees of freedom of the system. Space-time codes make it possible to take advantage of the gains made available by multiplexing (increasing the data rate and thus the coding rate of the system) and of the diversity of multi-antenna systems. By way of illustration, a multi-antenna system comprises Nt transmit antennas and Nr receive antennas. The coding rate is defined as r=n/T, where n is the number of information symbols sent during the time T. The maximum coding rate of a space-time code is equal to r=min(Nt,Nr). Let C be the following matrix of the space-time code of dimension Nt×T:
  C  =      [                                        c            11                                    …                                      c                          1              ⁢                                                          ⁢              T                                                            ⋮                          ⋱                          ⋮                                                  c                                          N                t                            ⁢              1                                                …                                      c                                          N                t                            ⁢              T                                            ]  
Each row i of the matrix feeds a transmit antenna i. Each element cij of the row i is a linear combination of information symbols and it is transmitted at instant jT. The pairwise error probability Pr{Z≠Z′} is defined as the probability that the receiver decodes the code word Z′≠Z when the code word Z was transmitted and the code word Z is a code word of the code C. This probability depends on two criteria relating to the structure of the code matrix C. Consider a matrix B of differences between Z and Z′:
  B  =      [                                                      z              11                        -                          z              11              ′                                                …                                                    z                              1                ⁢                                                                  ⁢                T                                      -                          z                              1                ⁢                                                                  ⁢                T                            ′                                                            ⋮                          ⋱                          ⋮                                                                z                                                N                  t                                ⁢                1                                      -                          z                                                N                  t                                ⁢                1                            ′                                                …                                                    z                                                N                  t                                ⁢                T                                      -                          z                                                N                  t                                ⁢                T                            ′                                            ]  
In reference [1] it is shown that in order to minimize the pairwise error probability, it is necessary for:
the rank of the matrix B.BH, for the set of all matrix pairs ZZ′, to be a maximum in order to guarantee maximum diversity. This defines the rank criterion of the code; and
the determinant of the matrix B.BH, for the set of all the matrix pairs ZZ′ to be maximized in order to minimize the pairwise error probability. This thus defines the determinant criterion of the code.
An example of a space-time code for Nt=2 and Nr=1 is described in Alamouti [2]. At a first instant, the symbols s1 and s2 are transmitted respectively by transmit antennas T1 and T2, and then at a second instant, the symbols −s*2 and s*1 are transmitted by the transmit antennas T1 and T2. In matrix form, the Alamouti code is represented in the following form:
      C    Alamouti    =      [                                        s            1                                                -                          s              2              *                                                                        s            2                                                s            1            *                                ]  
The Alamouti code makes it possible to achieve the maximum diversity equal to two since each symbol is transmitted in two independent versions (one per antenna). This observation is easily verified using the rank criterion. In addition, the code presents a full coding rate of one, since it makes it possible to send two symbols (n=2) during two symbol times (T=2).
So-called multiple-input multiple-output (MIMO) multi-antenna transmission systems provide space diversity, but the antennas need to be located at a common node of the network. Co-locating transmit and/or receive antennas does not give rise to difficulties with base stations. However, when the transmitters are terminals (mobile terminals in a cellular network, sensors in a network of sensors, . . . ), such co-location gives rise to constraints of size, of cost, and of hardware limitations.
So-called “co-operative” communications enable terminals having only one antenna to benefit from space diversity which is then referred to as “co-operation diversity”: single-antenna nodes of a multi-user network share their antennas so as to create a virtual multi-antenna system. Thus, such a new virtual MIMO system can exploit the known techniques for improving transmission quality that are used in MIMO systems. In co-operative systems with distributed transmission (e.g. using relays), the distributed antennas may be considered as being virtual multiple antennas that transmit to the destination. Consequently, the space-time codes can be used in a distributed manner.
Several earlier works address space-time coding for synchronous distributed systems. Under such circumstances, the frames are synchronized at relay level and at destination level. Nevertheless, given the imperfections of synchronization methods, the time delays due to multiple hops, the nature of the relays, cost, and the complexity of synchronization, synchronization is not always effective with antennas distributed over different nodes of the network. An absence of synchronization destroys synchronous code structures and leads to a loss of their properties (e.g. the rank criterion). By way of example, mention may be made of the Alamouti code. If the antenna T2 has a time delay of one symbol time relative to the antenna T1, then the code matrix perceived by the receiver becomes:
      [                                        s            1                                                -                          s              2              *                                                0                                      0                                      s            2                                                s            1            *                                ]     
Under such circumstances, the determinant is equal to: |ds1|4+2|ds1|2|ds2|2 which becomes zero (vanishes) if and only if the symbol s1 is the same for two different code words, dsi being the difference between the symbols at the position j for two code words.
In order to retain rank properties (which guarantee the maximum degree of diversity) and in order to retain the determinant of space-time codes when the signals received at the antennas are asynchronous, two main solutions have been proposed.
The first solution [3] describes designing asynchronous space-time codes that are robust against loss of synchronization and that reduce the cost of synchronization.
Consider two nodes T2 and 12, each having only one antenna that transmits to a destination in asynchronous manner, as shown in FIG. 1. Because of the distributed nature of the network, respective different time delays are introduced by the two nodes T2 and T2. The respective time delays at the destination D between the arrival of the signal coming from T1 and the arrival of the signal coming from T2 are written τ2 and τ2. The relative time delay between the nodes is written Δ=τ2−τ1. Without loss of generality, it can be considered that τ2≦τ1, so that Δ≧0. Without loss of generality, it can be considered that the relative time delay Δ is absorbed by the multiple-path effect and that it is an integer multiple of the symbol time. It is assumed that the time delays are known by the destination D but not necessarily by the nodes T1 and T2.
In the situation corresponding to FIG. 1, the authors of [3] have proposed a modified version of the Alamouti code in order to combat the effects of non-synchronization between the two nodes T1 and T2. The proposed code matrix is:
      C    Alamouti    D    =      [                                        s            1                                                -                          s              2              *                                                            -                          s              2              *                                                                        s            2                                                s            1            *                                                s            1            *                                ]  
If a time delay exists between receiving the two rows at the destination D, the determinant of the code matrix vanishes only if the two symbols s1 and s2 are the same for both code words; this new form of the Alamouti code can accommodate a lack of synchronization between the two transmitter nodes T2 and T2. Nevertheless, the fact of repeating the same symbols twice increases the size of the code and leads to a loss in its coding rate which becomes equal to r=2/3.
The second solution [4] recommends using the orthogonal frequency division multiplexing (OFDM) technique to eliminate the effect of time synchronization by using a cyclic prefix. Using the OFDM technique in that way serves to combat the effects of desynchronization in a wireless communications network by considering time delay as being due to multiple paths. Nevertheless, although that technique enables the frames from different antennas to be synchronized, it presents the following drawbacks:
at the outlet from the OFDM receiver, the signal transmitted by the two antennas has diversity equal to one even if it has taken two different paths;
in order to eliminate interference between the transmitted blocks, it is necessary to use a guard interval or a cyclic prefix that causes useful data rate to be lost and thus loses coding rate. In addition, in order to be certain to be able to synchronize the frames coming from antennas of the network, it is necessary for the guard interval or the cyclic prefix that is introduced to be of a length that is longer than the time delays between the antennas. For this purpose, it is necessary to increase the size of the guard interval or of the cyclic prefix so that it is greater than the maximum time delay that might be obtained between the antennas of the network; this also reduces coding rate; and
certain wireless communications systems do not make use of OFDM, and mention may be made for example of time-division multiple access (TDMA) or of code-division multiple access (CDMA) systems.